<p>数学系成立于2015年6月，现有基础数学、计算与应用数学、概率论与统计学3个学科方向，包含动力系统、代数、组合数学、数论、代数几何、偏微分方程、数学物理与微分几何、应用数学、计算数学、概率论、统计学、金融数学等12个主要研究领域。创系两年多，数学系于2018年1月成功获批数学学科的硕士、博士学位点。</p> //www.snoollab.com:80/handle/2SGJ60CL/35 2024-08-02T15:37:20Z 2024-08-02T15:37:20Z Guo, Sini Gu, Jia-Wen Ching, Wai-Ki Lyu, Benmeng //www.snoollab.com:80/handle/2SGJ60CL/789427 2024-08-01T04:10:56Z 2024-07-19T09:15:40Z

题名: Adaptive online mean-variance portfolio selection with transaction costs 作者: Guo, Sini; Gu, Jia-Wen; Ching, Wai-Ki; Lyu, Benmeng 摘要: Online portfolio selection is attracting increasing attention in both artificial intelligence and finance communities due to its efficiency and practicability in deriving optimal investment strategies in real investment activities where the market information is constantly renewed every second. The key issues in online portfolio selection include predicting the future returns of risky assets accurately given historical data and providing optimal investment strategies for investors in a short time. In the existing online portfolio selection studies, the historical return data of one risky asset is used to estimate its future return. In this paper, we incorporate the peer impact into the return prediction where the predicted return of one risky asset not only depends on its past return data but also the other risky assets in the financial market, which gives a more accurate prediction. An adaptive moving average method with peer impact (AOLPI) is proposed, in which the decaying factors can be adjusted automatically in the investment process. In addition, the adaptive mean-variance (AMV) model is firstly applied in online portfolio selection where the variance is employed to measure the investment risk and the covariance matrix can be linearly updated in the investment process. The adaptive online moving average mean-variance (AOLPIMV) algorithm is designed to provide flexible investment strategies for investors with different risk preferences. Finally, numerical experiments are presented to validate the effectiveness and advantages of AOLPIMV.

2024-07-19T09:15:40Z Liu, Xiaoyue Huang, Zhenzhong Song, Biwei Zhang, Zhen //www.snoollab.com:80/handle/2SGJ60CL/789342 2024-07-29T20:59:49Z 2024-07-19T08:43:50Z

题名: A LINEAR-PROGRAMMING PORTFOLIO OPTIMIZER TO MEAN-VARIANCE OPTIMIZATION 作者: Liu, Xiaoyue; Huang, Zhenzhong; Song, Biwei; Zhang, Zhen 摘要: In the Markowitz mean-variance portfolio optimization problem, the estimation of the inverse covariance matrix is not trivial and can even be intractable, especially when the dimension is very high. In this paper, we propose a linear-programming portfolio optimizer (LPO) to solve the Markowitz optimization problem in both low-dimensional and high-dimensional settings. Instead of directly estimating the inverse covariance matrix sigma-1, the LPO method estimates the portfolio weights sigma-1 mu through solving an l1-constrained optimization problem. Moreover, we further prove that the LPO estimator asymptotically yields the maximum expected return while preserving the risk constraint. To offer a practical insight into the LPO approach, we provide a comprehensive implementation procedure of estimating portfolio weights via the Dantzig selector with sequential optimization (DASSO) algorithm and selecting the sparsity parameter through cross-validation. Simulations on both synthetic data and empirical data from Fama-French and the Center for Research in Security Prices (CRSP) databases validate the performance of the proposed method in comparison with other existing proposals.

2024-07-19T08:43:50Z Guo, Ning Liu, Fei //www.snoollab.com:80/handle/2SGJ60CL/789334 2024-07-29T20:59:48Z 2024-07-19T08:42:14Z

题名: PURITY AND QUASI-SPLIT TORSORS OVER PRÃ FER BASES 作者: Guo, Ning; Liu, Fei 摘要: We establish an analogue of the Zariski-Nagata purity theorem for finite etale covers on smooth schemes over Prufer rings by demonstrating Auslander's flatness criterion in this non-Noetherian context. We derive an Auslander-Buchsbaum formula for general local rings, which provides a useful tool for studying the algebraic structures involved in our work. Through the analysis of reflexive sheaves, we prove various purity theorems for torsors under certain group algebraic spaces, such as the reductive ones. Specifically, using results from [EGA IV4] on parafactoriality on smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type groups at this level of generality. Subsequently, we leverage the aforementioned purity results to resolve the Grothendieck-Serre conjecture for torsors under a quasi -split reductive group scheme over schemes smooth over Prufer rings. Along the way, we also prove a version of the Nisnevich purity conjecture for quasi -split reductive group schemes in our Pruferian context, inspired by the recent work of Cesnavi & ccaron;ius [Ces22b].

2024-07-19T08:42:14Z Lu, Yanqing Zhang, Meng Tang, Ming //www.snoollab.com:80/handle/2SGJ60CL/789138 2024-08-01T04:24:18Z 2024-07-19T06:36:19Z

题名: Caching for Edge Inference at Scale: A Mean Field Multi-Agent Reinforcement Learning Approach 作者: Lu, Yanqing; Zhang, Meng; Tang, Ming 摘要: To enable AI-empowered Internet-of-things (AIoT) applications, it is crucial to achieve real-time data inference (e.g., prediction, control) at network edge. However, resource-constrained Internet-of-things devices (IoTDs) may be incapable of accomplishing those computation-intensive and latency-sensitive inference tasks. To address this issue, it is promising to incorporate mobile edge computing (MEC) systems and let IoTDs offload their inference tasks to edge servers that have already cached the associated neural network model required for inference. In this work, we take into account the limited storage and computing capacity of edge servers and formulate a neural network model caching problem for an MEC system with edge inference, in order to maximize the inference accuracy and reduce the task delay. To handle the exponential growth of signaling overhead and the learning difficulty under huge number of widely-deployed edge servers, we propose a cooperative mean field multi-agent reinforcement learning framework and a mean field actor-critic algorithm to solve the aforementioned problem. Simulation results show that our proposed algorithm outperforms several benchmarks, especially in large-scale edge networks.

2024-07-19T06:36:19Z Garoufalidis, Stavros Kashaev, Rinat //www.snoollab.com:80/handle/2SGJ60CL/789024 2024-08-01T04:22:34Z 2024-07-19T04:44:19Z

题名: The descendant colored Jones polynomials 作者: Garoufalidis, Stavros; Kashaev, Rinat 摘要: We discuss two realizations of the colored Jones polynomials of a knot, one appearing in an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another formulated in terms of a sequence of elements of the Habiro ring appearing in recent work of D. Zagier and the first author on the Refined Quantum Modularity Conjecture.

2024-07-19T04:44:19Z Fu, Zhaohui Tang, Tao Yang, Jiang //www.snoollab.com:80/handle/2SGJ60CL/789020 2024-08-01T04:26:47Z 2024-07-19T04:43:53Z

题名: ENERGY DIMINISHING IMPLICIT-EXPLICIT RUNGE-KUTTA METHODS FOR GRADIENT FLOWS 作者: Fu, Zhaohui; Tang, Tao; Yang, Jiang 摘要: This study focuses on the development and analysis of a group of high -order implicit-explicit (IMEX) Runge-Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time -step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high -order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high -order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.

2024-07-19T04:43:53Z Chen, Yanhong Jiang, Wenjun Zhang, Yiying //www.snoollab.com:80/handle/2SGJ60CL/788996 2024-07-29T20:58:00Z 2024-07-19T04:40:55Z

题名: Mean-variance insurance design under heterogeneous beliefs 作者: Chen, Yanhong; Jiang, Wenjun; Zhang, Yiying 摘要: In this paper we study an optimal insurance problem within the mean-variance framework for the case when the insured and insurer hold heterogeneous beliefs about the loss distribution. The implicit characterization of the optimal ceded loss function is obtained first, and we then parameterize the optimal ceded loss function in an explicit way for a general setup in which the insurer's belief Q either is or is not absolutely continuous with respect to the insured's belief P. We also show that the stop-loss function is optimal for the insured when the likelihood ratio function of the two parties' beliefs is decreasing. The connection between our framework and the expected utility framework is discussed. The situation where the insured can reduce both the mean and the variance of its loss through purchasing insurance is also investigated. Some analytical and numerical examples are presented to illustrate our results.

2024-07-19T04:40:55Z Guo, Hancheng Xiong, Jie Zheng, Jiayu //www.snoollab.com:80/handle/2SGJ60CL/788995 2024-07-29T20:58:00Z 2024-07-19T04:40:50Z

题名: Stochastic Maximum Principle for Generalized Mean-Field Delay Control Problem 作者: Guo, Hancheng; Xiong, Jie; Zheng, Jiayu 摘要: In this paper, we first derive the existence and uniqueness theorems for solutions to a class of generalized mean-field delay stochastic differential equations and mean-field anticipated backward stochastic differential equations (MFABSDEs). Then we study the stochastic maximum principle for generalized mean-field delay control problem. Since the state equation is distribution-depending, we define the adjoint equation as a MFABSDE in which all the derivatives of the coefficients are in Lions' sense. We also provide a sufficient condition for the optimality of the control.

2024-07-19T04:40:50Z Lv, Siyu Xiong, Jie //www.snoollab.com:80/handle/2SGJ60CL/788888 2024-08-01T04:33:01Z 2024-07-19T04:26:23Z

题名: A ZERO-SUM HYBRID STOCHASTIC DIFFERENTIAL GAME WITH SWITCHING CONTROLS 作者: Lv, Siyu; Xiong, Jie 摘要: In this paper, we study a zero-sum stochastic differential game in an infinite horizon, where the state equation consists of a number of diffusions coupled by a Markov chain and both players in the game employ switching controls. Based on the dynamic programming principle (DPP), the lower and upper value functions of the game are characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, which turns out to be two systems of variational inequalities with bilaterally inner-connected obstacles; thus the two value functions coincide and the game admits a value. Moreover, a verification theorem as a sufficient condition for Nash equilibriums is established, in which the equilibrium switching strategies for the two players are given in terms of the obstacle part of the HJBI equation.

2024-07-19T04:26:23Z Cui, Shumo Ding, Shengrong Wu, Kailiang //www.snoollab.com:80/handle/2SGJ60CL/788835 2024-08-01T04:30:28Z 2024-07-19T04:19:09Z

题名: ON OPTIMAL CELL AVERAGE DECOMPOSITION FOR HIGH-ORDER BOUND-PRESERVING SCHEMES OF HYPERBOLIC CONSERVATION LAWS 作者: Cui, Shumo; Ding, Shengrong; Wu, Kailiang 摘要: Cell average decomposition (CAD) plays a critical role in constructing boundpreserving (BP) high -order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant--Friedrichs--Lewy (CFL) condition is a fundamentally important yet difficult problem. The classic CAD, proposed in 2010 by Zhang and Shu using the Gauss-Lobatto quadrature, has been widely used over the past decade. Zhang and Shu only checked for the 1D \BbbP2 and \BbbP3 spaces that their classic CAD is optimal. However, we recently discovered that the classic CAD is generally not optimal for the multidimensional \BbbP2 and \BbbP3 spaces. Yet, it remained unknown for a decade what CAD is optimal for higher -degree polynomial spaces, especially in multiple dimensions. This paper presents the first systematical analysis and establishes the general theory on the OCAD problem, which lays a foundation for designing more efficient BP schemes. The analysis is very nontrivial and involves novel techniques from several branches of mathematics, including Carathe'\o dory's theorem from convex geometry, and the invariant theory of symmetric group in abstract algebra. Most notably, we discover that the OCAD problem is closely related to polynomial optimization of a positive linear functional on the positive polynomial cone, thereby establishing four useful criteria for examining the optimality of a feasible CAD. Using the established theory, we rigorously prove that the classic CAD is optimal for general 1D \BbbPk spaces and general 2D \BbbQk spaces of an arbitrary k \geq 1. For the widely used 2D \BbbPk spaces, the classic CAD is, however, not optimal, and we develop a generic approach to find out the genuine OCAD and propose a more practical quasi -optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD yet require much fewer nodes. These findings notably improve the efficiency of general high -order BP methods for a large class of hyperbolic equations while requiring only a minor adjustment of the implementation code. The notable advantages in efficiency are further confirmed by numerical results.

2024-07-19T04:19:09Z